Degrees of freedom can be described as the number of scores that
are free to vary. For example, suppose you tossed three dice. The total score
adds up to 12. If you rolled a 3 on the first die and a 5 on the second, then
you know that the third die must be a 4 (otherwise, the total would not add up
to 12). In this example, 2 die are free to vary while the third is not.
Therefore, there are 2 degrees of freedom.

In many situations, the degrees of freedom are equal to the
number of observations minus one. Thus, if the sample size were 20, there would
be 20 observations; the degrees of freedom would be 20 minus 1 or 19.

What is a chi-square critical value?

The chi-square critical value can be any number between
zero and plus infinity.
The chi-square calculator computes the probability that a chi-square statistic
falls between 0 and the critical value.

Suppose you randomly select a sample of 10
observations from a large population.
In this example, the degrees of freedom (DF) would be 9,
since DF = n - 1 = 10 - 1 = 9.
Suppose you wanted to find the probability that a chi-square statistic falls between
0 and 13. In the chi-square calculator, you would enter 9 for degrees of freedom
and 13 for the critical value. Then, after you click the Calculate button, the
calculator would show the cumulative probability to be 0.84.

What is a cumulative probability?

A cumulative probability is a sum of probabilities.
The chi-square calculator computes a cumulative probability. Specifically,
it computes the probability that a
chi-square statistic falls between 0 and some critical value (CV).

With respect to notation,
the cumulative probability that a chi-square statistic
falls between 0 and CV is indicated by P(Χ2< CV).

where σ is the standard deviation of
the population, s is the standard deviation of the sample, and n is the sample
size. The distribution of the chi-square statistic has n - 1 degrees of
freedom. (For more on the chi-square statistic, see the
tutorial on the chi-square distribution.)

What is a probability?

A probability is a number expressing the chances that a specific
event will occur. This number can take on any value from 0 to 1. A probability
of 0 means that there is zero chance that the event will occur; a probability
of 1 means that the event is certain to occur. Numbers between 0 and 1 quantify
the uncertainty associated with the event. For example, the probability of a
coin flip resulting in Heads (rather than Tails) would be 0.50. Fifty percent
of the time, the coin flip would result in Heads; and fifty percent of the
time, it would result in Tails.

where σ is the standard deviation of the population,
s is the standard deviation of the sample, and n is the sample size.

Now, using the Chi-Square Distribution
Calculator, we can determine the
cumulative probability for the chi-square statistic. We enter the
degrees of freedom (8) and the chi-square statistic (7.22) into the calculator,
and hit the Calculate button. The calculator reports that the cumulative
probability is 0.49. Therefore, there is a 49% chance that the sample
standard deviation will be no more than 0.95.

Find the chi-square critical value, if the cumulative
probability is 0.75 and the sample size is 25.

Given these data, we compute the chi-square statistic, using the
Chi-Square Distribution Calculator. We enter the degrees of freedom
(24) and the cumulative probability (0.75) into the calculator, and hit the
Calculate button. The calculator reports that the chi-square critical value is
28.2.

This means that if you select a random sample of 25 observations, there is
a 75% chance that the chi-square statistic from that sample will be less than
or equal to 28.2.